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I have encountered a problem regarding Heisenberg's uncertainty principle. Somewhere I found the expression

$$\Delta (x) \Delta (p) \geq \hbar$$

Whereas in some other places,

$$\Delta (x) \Delta(p) \geq \frac{h}{4\pi}$$

Here the second expression yields different expression than the first one. Why are the two expressions of uncertainty principle different? Which one is the correct form?

Qmechanic
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Lamichhane88
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  • Related http://physics.stackexchange.com/q/24116/ – Sanya Aug 01 '16 at 16:54
  • This principle is almost never used for exact calculations but rather to estimate orders of magnitude so you can just as well write $\Delta x \Delta o \geq \pi^{0.3} \hbar$ and it will still be valid. – Blazej Aug 01 '16 at 22:07
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    Possible duplicates: http://physics.stackexchange.com/q/69604/2451 , http://physics.stackexchange.com/q/103208/2451 and links therein. – Qmechanic Aug 02 '16 at 09:31

1 Answers1

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See Ballentine: Quantum mechanics p. 224 (chapter 8): the uncertainty principle applied to position and momentum yields, in 1 dimension: $$ \Delta_X \Delta_P \geq \frac{\hbar}{2}=\frac{h}{4 \pi} $$ But the other version is often used when it's mainly about an estimation of magnitude and the factor of $0.5$ does not really matter.

Edit: On Wikipedia it says that the first unproofed heuristically motivated version of the principle given by Heisenberg himself was without the factor of $0.5$. Maybe that is a historical explanation.

Sanya
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